Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 18F (Galois Theory)

The squared Vandermonde product defining the polynomial discriminant is symmetric in all roots. By the fundamental theorem of symmetric polynomials, it is a polynomial in the elementary symmetric functions, which are the signed coefficients of the monic polynomial. Hence it is a polynomial function of those coefficients over the base field.

For $f=X^3+pX+q$, $\Delta (f)=-\mathrm{Res}(f,f^{\prime })$. Eliminating the two leading rows in the Sylvester determinant reduces the resultant toThusFor $X^3-3X+1$, the only possible rational roots are $\pm 1$, and neither is a root. The cubic is therefore irreducible, so its Galois group is a transitive subgroup of $S_3$. Its discriminant is $81=9^2$. The Vandermonde product changes sign under odd permutations, and its square being a rational square implies that the product itself is rational; hence all Galois permutations are even. The only transitive subgroup of $A_3$ is $A_3$ itself. Consequently, writing $L$ for its splitting field,